1 Multilevel polynomial regression analysis

Multilevel polynomial regression analysis (MPRA; see Nestler et al. (2019)) is an adaptation of polynomial regression analysis (PRA; see Edwards & Parry (1993)) that can be used when data have a multilevel structure (e.g., observations across multiple time points nested within participants). In the current study, MPRA allows testing and interpreting both the linear effects of vitality and learning separately, as well as the effects of congruent vitality and learning scores using data collected at four time points with a one-month time lag between the assessments. In the following, we use the index \({m}\) to denote the Level 2 unit (\(m = 1…N\)). To assess the existence of congruence effects, the following parameters are estimated at the population level: the slope of the line of congruence (LOC) (i.e., vitality = learning) as related to the outcome variable is given by \(\alpha_{1m} = b_{1} + b_{2}\). Here, \(b_{1}\) is the unstandardized regression coefficient for the (latent) vitality variable and \(b_2\) is the regression coefficient for the (latent) learning variable. The curvature along the line of perfect agreement is specified by \(\alpha_{1m} = b_{3} + b_{4} + b_{5}\). Here, \(b_{3}\) is the regression coefficient for (latent) vitality squared, b4 is the regression coefficient for the cross-product of the (latent) vitality and the (latent) learning variable, and b5 is the unstandardized regression coefficient for latent learning squared. The slope of the line of incongruence (LOIC) is defined as \(\alpha_{3m} = b_{1} - b_{2}\). The curvature of the line of incongruent vitality and learning levels as related to the outcome variable is specified as \(\alpha_{4m} = b_{3} - b_{4} + b_{5}\). Each of the regression coefficients \(b_{1}\) to \(b_{5}\) includes a fixed effect and Level 2 residual terms. The fixed effects of the regression coefficients are used to estimate the average response surface parameters \(\hat{\alpha}_{1}\) to \(\hat{\alpha}_{5}\) (Nestler et al., 2019).

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References

Edwards, J. R., & Parry, M. E. (1993). On the use of polynomial regression equations as an alternative to difference scores in organizational research. Academy of Management Journal, 36(6), 1577–1613.
Nestler, S., Humberg, S., & Schönbrodt, F. D. (2019). Response surface analysis with multilevel data: Illustration for the case of congruence hypotheses. Psychological Methods, 24(3), 291–308. https://doi.org/10.1037/met0000199

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2 Time-lagged correlations

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3 Interactive response surface plots

3.1 Physical health

The dots represent the raw data points. The x and y axes represent the person-mean centred vitality and learning scores, respectively.

3.2 Mental health

The dots represent the raw data points. The x and y axes represent the person-mean centred vitality and learning scores, respectively.

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4 Multilevel lagged analyses

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4.1 Measurement invariance

N = 1,064. CFI = comparative fit index; TLI = Tucker–Lewis index; RMSEA = root mean square error of approximation; SRMR = standardized root mean square residual.
Model test \(\chi^2\) \(\textit{p}\) CFI TLI RMSEA SRMR \(\Delta\chi^2\) \(\Delta\)CFI \(\Delta\)TLI \(\Delta\)RMSEA \(\Delta\)SRMR
Configural invariance 404.625 (188) <.001 0.990 0.985 0.033 0.032 - - - - -
Weak invariance 412.586 (200) <.001 0.990 0.986 0.032 0.032 7.961 (12) <0.001 0.001 -0.001 <0.001
Strong invariance 434.928 (212) <.001 0.990 0.986 0.031 0.032 22.342 (12) <0.001 <0.001 <0.001 <0.001
Strict invariance 445.956 (218) <.001 0.989 0.986 0.031 0.033 11.028 (6) <0.001 <0.001 <0.001 <0.001

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4.2 Physical health

Lagged multilevel model including the effects of predictors measured at t-1 on physical health measured at t

Model 1:Physical health predicted by thriving and control variables; Model 2: Physical health predicted by learning and vitality linear, interaction, and squared terms and control variables.
  Model 1 Model 2
Predictors b SE beta SE beta p b SE beta SE beta p std. p
(Intercept) 5.40 0.17 -0.00 0.03 <0.001 5.62 0.29 0.00 0.03 <0.001 0.967
T-1.Physical.health -0.25 0.03 -0.12 0.01 <0.001 -0.25 0.03 -0.12 0.01 <0.001 <0.001
Thriving 0.01 0.04 0.00 0.02 0.766
Vitality -0.01 0.03 -0.01 0.02 0.734 0.734
Learning 0.02 0.03 0.01 0.01 0.350 0.350
Vitality squared -0.02 0.02 -0.01 0.02 0.459 0.459
Vitality x learning 0.00 0.02 0.00 0.02 0.855 0.855
Learning squared 0.02 0.02 0.02 0.02 0.319 0.319
Positive affect -0.01 0.04 -0.01 0.02 0.704 -0.01 0.04 -0.00 0.02 0.780 0.780
Negative affect -0.02 0.03 -0.01 0.01 0.582 -0.02 0.03 -0.01 0.01 0.596 0.596
Demands -0.00 0.03 -0.00 0.01 0.971 -0.00 0.03 -0.00 0.01 0.979 0.979
Autonomy 0.02 0.03 0.01 0.01 0.425 0.02 0.03 0.01 0.01 0.428 0.428
Coworker support -0.01 0.03 -0.01 0.02 0.638 -0.01 0.03 -0.01 0.02 0.702 0.702
Supervisor support 0.00 0.03 0.00 0.02 0.914 -0.00 0.03 -0.00 0.02 0.997 0.997
Time 0.01 0.01 0.01 0.01 0.470 0.01 0.01 0.01 0.01 0.503 0.503
Mean vitality -0.30 0.16 -0.32 0.17 0.063 0.063
Mean learning 0.14 0.17 0.14 0.17 0.409 0.409
Mean vitality squared 0.06 0.03 0.41 0.20 0.038 0.038
Mean vitality x learning -0.02 0.05 -0.09 0.25 0.713 0.713
Mean learning squared -0.02 0.03 -0.12 0.20 0.566 0.566
Mean thriving -0.01 0.06 -0.01 0.05 0.853
Mean positive affect 0.01 0.06 0.01 0.06 0.883 -0.01 0.07 -0.01 0.06 0.920 0.920
Mean negative affect -0.31 0.04 -0.26 0.03 <0.001 -0.30 0.04 -0.26 0.03 <0.001 <0.001
Mean demands -0.06 0.03 -0.06 0.03 0.046 -0.06 0.03 -0.05 0.03 0.075 0.075
Mean autonomy 0.07 0.03 0.07 0.03 0.047 0.06 0.03 0.06 0.03 0.088 0.088
Mean coworker support 0.01 0.04 0.01 0.04 0.723 0.01 0.04 0.01 0.04 0.777 0.777
Mean supervisor support 0.01 0.04 0.01 0.04 0.823 0.01 0.04 0.01 0.04 0.859 0.859
Age -0.01 0.00 -0.16 0.03 <0.001 -0.01 0.00 -0.17 0.03 <0.001 <0.001
Gender -0.03 0.05 -0.02 0.03 0.492 -0.04 0.05 -0.02 0.03 0.435 0.435
Education 0.08 0.03 0.08 0.03 0.006 0.08 0.03 0.08 0.03 0.007 0.007
Organizational tenure 0.01 0.00 0.06 0.03 0.073 0.01 0.00 0.07 0.03 0.048 0.048
Random Effects
σ2 0.256 0.250
τ00 0.398 Person.ID 0.397 Person.ID
τ11 0.025 Person.ID.Thriving 0.039 Person.ID.Vitality
ρ01 -0.118 Person.ID -0.033 Person.ID
ICC 0.611 0.618
N 888 Person.ID 888 Person.ID
Observations 2136 2136
Marginal R2 / Conditional R2 0.123 / 0.658 0.128 / 0.667

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4.3 Mental health

Lagged multilevel model including the effects of predictors measured at t-1 on mental health measured at t

Model 1:Mental health predicted by thriving and control variables; Model 2: Mental health predicted by learning and vitality linear, interaction, and squared terms and control variables.
  Model 1 Model 2
Predictors b SE beta SE beta p b SE beta SE beta p
(Intercept) 4.39 0.12 0.00 0.02 <0.001 4.14 0.20 0.00 0.02 <0.001
T-1.Mental.health -0.37 0.03 -0.16 0.01 <0.001 -0.37 0.03 -0.17 0.01 <0.001
Thriving 0.05 0.04 0.02 0.02 0.154
Vitality 0.00 0.03 0.00 0.02 0.941
Learning 0.06 0.03 0.03 0.01 0.020
Vitality squared -0.03 0.02 -0.02 0.01 0.191
Vitality x learning 0.02 0.02 0.01 0.02 0.361
Learning squared 0.01 0.02 0.01 0.01 0.577
Positive affect -0.00 0.04 -0.00 0.01 0.947 0.00 0.04 0.00 0.01 0.985
Negative affect -0.05 0.03 -0.02 0.01 0.188 -0.05 0.03 -0.02 0.01 0.166
Demands -0.03 0.03 -0.01 0.01 0.270 -0.03 0.03 -0.01 0.01 0.271
Autonomy 0.01 0.03 0.00 0.01 0.749 0.01 0.03 0.01 0.01 0.619
Coworker support 0.02 0.03 0.01 0.01 0.586 0.01 0.03 0.01 0.01 0.664
Supervisor support -0.02 0.03 -0.01 0.01 0.500 -0.02 0.03 -0.01 0.01 0.519
Time 0.00 0.01 0.00 0.01 0.752 0.01 0.01 0.00 0.01 0.695
Mean vitality 0.61 0.11 0.56 0.10 <0.001
Mean learning -0.30 0.12 -0.26 0.10 0.012
Mean vitality squared -0.06 0.02 -0.34 0.12 0.005
Mean vitality x learning -0.02 0.03 -0.11 0.15 0.463
Mean learning squared 0.05 0.02 0.27 0.13 0.032
Mean thriving 0.03 0.04 0.03 0.03 0.440
Mean positive affect 0.42 0.04 0.35 0.03 <0.001 0.30 0.05 0.25 0.04 <0.001
Mean negative affect -0.67 0.03 -0.50 0.02 <0.001 -0.65 0.03 -0.48 0.02 <0.001
Mean demands -0.02 0.02 -0.01 0.02 0.452 -0.00 0.02 -0.00 0.02 0.983
Mean autonomy 0.01 0.02 0.01 0.02 0.576 0.03 0.02 0.02 0.02 0.268
Mean coworker support 0.07 0.03 0.06 0.02 0.021 0.06 0.03 0.05 0.02 0.025
Mean supervisor support -0.06 0.03 -0.05 0.03 0.031 -0.07 0.03 -0.07 0.02 0.006
Age 0.01 0.00 0.06 0.02 0.003 0.01 0.00 0.06 0.02 0.003
Gender -0.01 0.04 -0.01 0.02 0.732 -0.01 0.03 -0.00 0.02 0.838
Education -0.04 0.02 -0.04 0.02 0.037 -0.04 0.02 -0.03 0.02 0.042
Organizational tenure -0.00 0.00 -0.01 0.02 0.744 -0.00 0.00 -0.01 0.02 0.525
Random Effects
σ2 0.266 0.273
τ00 0.138 Person.ID 0.123 Person.ID
τ11 0.079 Person.ID.Thriving 0.036 Person.ID.Vitality
ρ01 -0.053 Person.ID 0.017 Person.ID
ICC 0.360 0.323
N 888 Person.ID 888 Person.ID
Observations 2136 2136
Marginal R2 / Conditional R2 0.580 / 0.731 0.591 / 0.723